13 papers
Essential-Surface Complexes of Knot Exteriors: Image, Kernel, and Reconstruction
Makoto Ozawa
The curve complex of a surface is deliberately forgetful, yet in most cases its automorphism group recovers the mapping class group. We develop an analogous image--kernel--reconstr…
A Geometric Finiteness Theory for Essential Surfaces in Knot Exteriors
Makoto Ozawa
We develop a relative geometric finiteness theory for essential surfaces in knot exteriors. Let be a unit-thickness representative of a knot type , with $\operatorname{Len}…
Density and Compression on Lattice-Knot Merge Trees
Makoto Ozawa
Ropelength is usually studied as a minimization problem for one knot at a time. We instead use ropelength to filter the space of all realizations of a knot type. For a knot type $K…
Finite Knot Theory via Ropelength-Filtered Reidemeister Graphs
Makoto Ozawa
We develop a finite-recognition framework for knot types using ropelength-filtered Reidemeister graphs. For a fixed projection direction, bounded-ropelength thick-knot spaces deter…
Geometric densities and compression radii of knot types
Makoto Ozawa
We introduce scale-free compression radii and packing ratios of knot types and clarify their relation to geometric densities. Let be a Euclidean-invariant, scale-covariant size…
Merge Trees of Lattice Knots
Makoto Ozawa
We study length-filtered move graphs of lattice knots as finite-state models for quantitative reconfiguration. At level , vertices are lattice-polygon representatives of a fixed…